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The 3-dimensional PPNG (pure proportional navigation guidance) law was proposed about forty years ago, but its performance has been analyzed only on the basis of the linearized trajectory equations, which are valid locally around the collision course. We take into full account the nonlinear dynamic characteristics of pursuit situation in the 3-dimensional space to analyze the performance of the 3-dimensional PPNG law more rigorously. We prove that a missile guided by the 3-dimensional PPNG law can always intercept a target maneuvering randomly in the 3-dimensional space if 1) the target acceleration varies with a known bound, 2) the navigation constant is selected large, and either 3a) the initial heading error is small or 3b) the missile keeps the head toward the target during flight. We also propose a modified PNG law, which seems to exhibit better performance at the final phase of pursuit than the conventional PPNG law. We introduce a Lyapunov-like method that proves to be a very powerful tool in obtaining our results.>
Song et al. (Sat,) studied this question.